Sometimes you can substitute values for $n_0$ and $c$ in the big-$O$ equation and compare two functions. Or take limits and compare two functions.

But for the following functions, for example, taking the limit in infinity for

$f_3$ over $f_2$ requires using l’HÃ´pital’s rule which doesn’t simplify anything. $f_3$ is technically the product of a polynomial and an exponential function. And I don’t know how to go with comparing functions like that with others.

Firstly, I know that $f_4$ is the most efficient because it is $O(n^2)$. ($f_4(n) = n + frac{n(n + 1)}{2}$) and the rest are exponential.

But for the rest, I really don’t know what to besides using my intuition which could be really far from the correct answer anyway. Please help me compare these rigorously.

$f_1(n) = n^{sqrt{n}}$

$f_2(n) = 2^n$

$f_3(n) = n^{100}2^{frac{n}{2}}$

$f_4(n) = Sigma_{i=1}^{n}i + 1$